ABSTRACT Let or . Let be a connected linear algebraic group over . Under the assumption that the -variety is -rational, i.e. that the function field is purely transcendental, it was proved that a principal homogeneous space of has a rational point over as soon as it has one over each completion of with respect to a valuation. In this paper we show that one cannot in general do without the -rationality assumption. To produce our examples, we introduce a new type of obstruction. It is based on higher reciprocity laws on a 2-dimensional scheme. We also produce a family of principal homogeneous spaces for which the refined obstruction controls exactly the existence of rational points.